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The polynomial built from derivatives of a function at a point.
A Taylor polynomial of order for a function at a point is the polynomial , assuming the needed higher derivatives exist at .
Taylor polynomials package derivative data at a point into a single polynomial that locally approximates . The approximation is quantified by Taylor's theorem with remainder.
A higher derivative of a function is a derivative of order , defined recursively by , , and wherever the derivative exists.
A polynomial (in one real variable) is a function of the form
for some integer and coefficients . If , the integer is the degree of .
Taylor's theorem with remainder: Let be an interval, let , and let have continuous derivatives up to order on . For each there exists a point between and such that
The polynomial part is the Taylor polynomial of degree at , and the final term is the Lagrange-form remainder, which provides effective error bounds when is bounded.